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Research
Report
Series
On Coloring the Rational
Quantum Sphere
Hans Havlicek
Institut für Geometrie, Technische
Universität Wien, Austria
Günther Krenn
Johann Summhammer
Atominstitut der Österreichishen
Universitäten, Austria
Karl Svozil
Theoretische Physik, Technische
Universität Wien, Viena Austria
CDMTCS-121
February 2000
Centre for Discrete Mathematics and
Theoretical Computer Science
On coloring the rational quantum sphere
Hans Havlicek
Institut fur Geometrie, Technische Universitat Wien
Wiedner Hauptstrae 8-10/1133, A-1040 Vienna, Austria
[email protected]
and
Gunther Krenn and Johann Summhammer
Atominstitut der osterreichischen Universitaten,
Stadionallee 2
A-1020 Vienna, Austria
[email protected], [email protected]
and
Karl Svozil
Institut fur Theoretische Physik, Technische Universitat Wien
Wiedner Hauptstrae 8-10/136, A-1040 Vienna, Austria
e-mail: [email protected]
(to whom correspondence should be directed)
http://tph.tuwien.ac.at/e svozil/publ/coloring.fhtm,ps,texg
Abstract
We discuss types of colorings of the rational quantum sphere similar to
the one suggested recently by Meyer [1], in particular the consequences for
the Kochen-Specker theorem and for the correlation functions of entangled
subsystems.
Recently, Godsil and Zaks [2] published a constructive coloring of the rational
unit sphere with the property that for any orthogonal tripod formed by rays
extending from the origin of the points of the sphere, exactly one ray is red,
white and black. They also showed that any consistent coloring of the real
sphere requires an additional color.
Based on this very interesting result, Meyer [1] suggested that the physical
impact of the Kochen-Specker theorem [3] is \nullied," since for all practical
purposes it is impossible to operationalize the dierence between any dense set
of rays and the continuum of Hilbert space rays.
We shall argue here that Meyer's result is itself \nullied" for a formal and
for a physical reason: (i) the non-closedness of the resulting set of propositions
1
under quantum logical operations; in particular under the nor-operation; (ii) the
continuity of Bell-type two-particle correlation function rules out the possibility
that \unperformed experiments have results" [4].
In what follows we shall consider \rational rays." A \rational ray" is the
linear span of a non-zero vector of Q n Rn .
Let p be a prime number. A coloring of the rational rays of Rn , n 1, using
n
;
p 1 + pn;2 + : : : + 1 colors can be constructed in a straightforward manner.
We refer to [5, 6, 7] for the theoretical background of the following construction.
Each rational ray is the linear span of a vector (x1 ; x2 ; : : : ; xn ) 2 Zn, where
x1 ; x2 ; : : : ; xn are coprime. Such a vector is unique up to a factor 1.
Next, let Zp be the eld of residue classes modulo p. The vector space Znp
has pn ; 1 non-zero vectors; each ray through the origin of Znp has p ; 1 non-zero
vectors. So there are exactly (pn ; 1)=(p ; 1) = pn;1 + pn;2 + : : : + 1 distinct
rays through the origin which can be colored with pn;1 + pn;2 + : : : + 1 distinct
colors.
Finally, assign to the ray Sp(x1 ; x2 ; : : : ; xn ) (\Sp" denotes linear span) the
color of the ray of Znp which is obtained by taking the modulus of the coprime integers x1 ; x2 ; : : : ; xn modulo p. Observe that x1 ; x2 ; : : : ; xn cannot vanish simultaneously modulo p and that (x1 ; x2 ; : : : ; xn ) yield the same color. Obviously,
all pn;1 + pn;2 + : : : + 1 colors are actually used.
In what follows, we consider the case p = 2, n = 3. Here all rational rays
Sp(x; y; z ) (with x; y; z 2 Z coprime) are colored according to the property which
ones of the components x; y; z are even (E) and odd (O). There are exactly 7
of such triples OEE, EOE, EEO, OOE, EOO, OEO, OOO which are associated
with one of seven dierent colors #1; #2; #3; #4; #5; #6; #7. Only the EEE
triple is excluded. Those seven colors can be identied with the seven points of
the projective plane over Z2; cf. Fig. 1.
Next, we restrict our attention to those rays which meet the rational unit
sphere S 2 \ Q 3 . The following statements on a triple (x; y; z ) 2 Z3 n f(0; 0; 0)g
(not necessarily coprime) are equivalent:
(i) The ray Sp(x; y; z ) intersects the unit sphere
p at two rational points; i.e., it
contains the rational points (x; y; z ) = x2 + y2 + z 2 2 S 2 \ Q 3 .
(ii) The Pythagorean property holds, i.e., x2 + y2 + z 2 = n2 ; n 2 N .
This equivalence can be demonstrated
as
follows. All points on the rational unit
;
sphere can be written as r = aa ; bb ; cc with a; b; c 2 Z, a0 ; b0 ; c0 2 Z n f0g, and
; a 2 + ; b 2 + ; c 2 = 1. Multiplication of r with a02b02c02 results in a vector
a
b
c
of Z3 satisfying (ii). ;Conversely,
from x2 + y2 + z 2 = n2 ; n 2 N , we obtain the
y z
x
rational unit vector n ; n ; n 2 S 2 \ Q 3 .
Notice that this Pythagorean property is rather restrictive. Not all rational
rays intersect the rational unit sphere.
p For a proof, consider Sp(1; 1; 0) which
intersects the unit sphere at (1= 2)(1; 1; 0) 62 S 2 \ Q 3 . Although both the set
0
0
0
0
0
0
2
EEO
v
#3
T
dv
T
T
T
OEO #1
v
OEE
v
v
v
T EOO
T
T
OOO T
T
T
T
T
OOE
v #2
EOE
Figure 1: The projective plane over Z2. and the reduced coloring scheme discussed.
of rational rays as well as S 2 \ Q 3 are dense, there are \many" rational rays
which do not have the Pythagorean property.
If x; y; z are chosen coprime then a necessary condition for x2 + y2 + z 2 being
a non-zero square is that precisely one of x, y, and z is odd. This is a direct
consequence of the observation that any square is congruent to 0 or 1, modulo
4, and from the fact that at least one of x, y, and z is odd. Hence our coloring
of the rational rays induces the following coloring of the rational unit sphere
with those three colors that are represented by the standard basis of Z32:
color #1 if x is odd, y and z are even,
color #2 if y is odd, z and x are even,
color #3 if z is odd, x and y are even.
All three colors occur, since the vectors of the standard basis of R3 are
colored dierently.
Suppose that two points of S 2 \ Q 3 are on rays Sp(x; y; z ) and Sp(x0 ; y0; z 0 ),
each with coprime entries. The inner product xx0 + yy0 + zz 0 is even if and only
if the inner product of the corresponding basis vectors of Z32 is zero or, in other
words, the points are colored dierently. In particular, three points of S 2 \ Q 3
with mutually orthogonal position vectors are colored dierently.
From our considerations above, three colors are sucient to obtain a coloring
of the rational unit sphere S 2 \ Q 3 such that points with orthogonal position
vectors are colored dierently, but clearly this cannot be accomplished with two
3
colors. So the \chromatic number" for the rational unit sphere is three. This
result is due to Godsil and Zaks [2]; they also showed that the chromatic number
of the real unit sphere is four. However, they obtained their result in a slightly
dierent way. Following [6] all rational rays are associated with three colors by
making the following identication:
#1;
#2 = #4;
#3 = #5 = #6 = #7:
This 3-coloring has the property that coplanar rays are always colored by using
only two colors; cf. Fig. 1. According to our approach this intermediate 3coloring is not necessary, since rays in colors #4, #5, #6, #7 do not meet the
rational unit sphere.
As a corollary, the rational unit sphere can be colored by two colors such
that, for any arbitrary orthogonal tripod spanned by rays through its origin,
one vector is colored by color #1 and the other two rays are colored by color
#2. This can be easily veried by identifying colors #2 & #3 from the above
scheme. (Two equivalent two-coloring schemes result from a reduced chromatic
three-coloring scheme by requiring that color #1 is associated with x or y being
odd, respectively.)
It can also be shown that each color class in the above coloring schemes
is dense in the sphere. To prove this, Godsil and Zaks consider such that
sin = 53 and thus cos = 54 . is not a rational multiple of ; hence sin(n)
and cos(n) are non-zero for all integers n. Let F be the rotation matrix about
the z -axis through an angle ; i.e.,
0 cos F = @ ; sin 0
sin 0
cos 0
0 1
1
A:
Then the image I , under the powers of F , of the point (1; 0; 0) is a dense subset
of the equator.
; Now suppose that the point u = ac ; cb ; is on the rational unit sphere and
that a; c are odd and thus b is even. In the coloring scheme introduced above,
u has the same color as (1; 0; 0) (identify a = c = 1 and b = 0); and so does F u.
This proves that I (the image of all powers of F of the points u) is dense. We
shall come back to the physical consequences of this property later.
In the reduced two-color setting, if the two "poles" (0; 0; 1) acquire color
#1, then the entire equator acquires color #2. Thus, for example, for the two
tripods spanned by f(1; 0; 0); (0; 1; 0); (0; 0; 1)g and f(3; 4; 0); (;4; 3; 0); (0; 0; 1)g,
the rst two legs have color #2, while (0; 0; 1) has color #1.
The above coloring scheme of the rays through the origin meeting the rational sphere is not closed under certain geometrical operations such as taking
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two orthogonal ray of the subspace spanned by two non-collinear rays (the cross
product of the associated vectors). This can be easily demonstrated by considering the two vectors
3 4 4 3
2
3
5 ; 5 ; 0 ; 0; 5 ; 5 2 S \ Q :
The cross product thereof is
12 ;9 12 2
3
25 ; 25 ; 25 62 S \ Q :
Indeed, if instead of S 2 \ Q 3 we would start with three non-orthogonal, noncollinear rational rays and generate new ones by the cross product, we would
end up with all rational rays [8].
From a quantum logic point of view, this non-closedness under elementary
operations such as the nor-operation might be considered a serious deciency
which rules out the above model as an alternative candidate for Hilbert space
quantum mechanics. Indeed, it is just the relative (with respect to other sets
such as the rational rays) \thinness" which guarantees colorability, but which on
the other hand does not allow closedness under the quantum logical operations.
It may, however, be argued that the non-closedness is among counterfactual,
non-commuting properties which have no direct operational meaning. But then
it would be entirely senseless to consider any but six points of the rational sphere
corresponding to intersections with a single tripod (that one being measured),
which would make any coloring trivial.
Of course, this leaves open the question as to whether or not there exist
dense sets of chromatic number three which are closed under quantum logical
operations, in particular under the nor-operation.
The coloring schemes discussed above have a physical interpretation as follows. Any linear subspace Sp r of a vector r can be identied with the associated
projection operator Er and with the quantum mechanical proposition \the physical system is in a pure state Er " [9]. The coloring of the associated point on the
unit sphere (if it exists) is equivalent with a valuation or two-valued probability
measure
P r : Er 7! f0; 1g
where 0 #2 and 1 #1. That is, the two colors #1, #2 can be identied
with the classical truth values and with the EPR- type \elements of physical
reality" [10]:
\It is true that the physical system is in a pure state Er ."
\It is false that the physical system is in a pure state Er ."
Since, as has been argued before, the rational unit sphere has chromatic
number three, two colors suce for a reduced coloring generated under the
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assumption that the colors of two rays in any orthogonal tripod are identical.
This eectively generates consistent valuations or \elements of physical reality"
associated with the dense subset of physical properties corresponding to the
rational unit sphere [1]. For an extension of these arguments, see Kent [11] and
Clifton and Kent [12].
We may indeed take for granted that for all practical (operational) physical
purposes, a dense subset cannot be distinguished from the usual Hilbert space
based upon real or complex number elds. This is, at least to our knowledge, one
of the very rare instances where it does make a dierence whether we consider a
real rational quantity or merely its rational double. The two cases lie on dierent
sides of a demarcation line of classicality and (maybe) an \understanding" of
quantum mechanics.
To put it dierently: given any nonzero measurement uncertainty " and any
non-colorable Kochen-Specker graph ;(0) [3, 13], there exists another Graph
(in fact, a denumerable innity thereof) ;() which lies inside the range of
measurement uncertainty " [and thus cannot be discriminated from the
non-colorable ;(0)] which can be colored. Such a graph, however, might not be
connected in the sense that the associated subspaces can be cyclically rotated
into itself by local transformation along single axes. The set ;() might thus
correspond to a collection of tripods such that none of the axes coincides with
any other, although all of those non-identical single axes are located within apart from each other.
If this is indeed the case, then the reason why a Kochen-Specker type contradiction does not occur is the impossibility to \close" the argument; to complete
the cycle: the necessary elements of physical reality are simply not available in
the rational sphere model. (For the same reason, an equilateral triangle does
not exist in Q 2 [14].)
However, in addition to the non-closedness under elementary quantum logical operations we would like to point out another, rather serious problem. Due
to the density of points colored with any single color, any such coloring is noncontinuous in the sense that between any two points of equal color there is a
point (indeed, an innity thereof) of dierent color.
Let us then assume that any such value-deniteness might apply also to
non-local setups and consider Bell-type correlation functions for spin- 21 state
measurements. For singlet states along two directions which are an angle apart, the quantum probabilities to nd identical two particle states ++ or ;;
is P = = P ++ + P ;; = sin2 (=2), whereas for the non-identical states +; and
;+ it is P 6= = P +; + P ;+ = cos2 (=2). The corresponding classical quantities
are P = = = and P 6= = 1 ; = [15, 16, 17]. In the quantum case, a statistical
argument [4, 16, 17] demonstrates that \elements of physical reality" do not
exist, whereas in the classical case they can be dened.
Thus if there exists any coloring of the associated fourdimensional model,
any such coloring should be in accordance with the physical ndings which
support quantum mechanics. In particular, any probability theory derived from
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the valuations of dense subsets of Hilbert spaces should be able to reproduce
the well-known quantum correlations which violate value-deniteness as well
as locality. This, however, is neither possible with the usual classical valuedeniteness, nor with the discontinuity originating in the denseness of any single
color.
Acknowledgments
The authors would like to acknowledge stimulating discussions with Ernst Specker.
References
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http://xxx.lanl.gov/abs/quant-ph/9905080.
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